= Johnson-Segalman negative-slope shear threshold
{c}
{title2=$\mu_0/(\alpha^2G_0\tau)<1/8$}
For nonzero slip parameter with $|\alpha|<1$, scale shear rate as $x=\tau\sqrt{1-\alpha^2}\dot\gamma$ and stress as $S=x[b+1/(1+x^2)]$, where $b=\mu_0/(\alpha^2G_0\tau)$. The derivative is $b+(1-x^2)/(1+x^2)^2$ and its least value is $b-1/8$, attained at $x^2=3$. A decreasing homogeneous constitutive segment therefore exists exactly under the displayed condition. It permits different shear rates at the same stress and motivates <shear banding>; the local curve alone does not select a band-interface stress.
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